#499684

order	512
nilpotency class	4
isoclinism	isoclinic class of #498968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#486
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23815
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(3,14,17,18)(4,19)(6,20)(9,24,26,28)(10,12)(15,29,30,25)(16,31,32,22)(21,27);(1,2)(3,15,17,30)(4,10)(5,11)(6,21)(7,23)(8,13)(9,16,26,32)(12,19)(14,25,18,29)(20,27)(22,28,31,24);(1,3)(2,9)(4,16)(5,18)(6,22)(7,14)(8,17)(10,25)(11,28)(12,29)(13,26)(15,27)(19,32)(20,31)(21,30)(23,24);(1,4,5,20,8,19,7,6)(2,10,11,27,13,12,23,21)(3,16,18,31,17,32,14,22)(9,25,28,15,26,29,24,30)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ef,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=ghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406880);